Statistical Foundations of Insurance Risk Pooling

1
Statistical Foundations of Insurance(Risk
Pooling)

• Lecture 3

2
Random Variables

• Variables can be either deterministic or random
• The outcome of a deterministic variable is known
• The outcome from a random variable is unknown

3
Probability Distribution

• Identifies all the possible outcomes and the
  probabilities of the outcomes
• Examples
• Tossing a coin
• Throwing two dice

4
Probability Distribution for a Coin Toss
Prob
.50
Heads
Tails
5
Probability Distribution for Thrown Dice
Prob
6/36
3/36
1/36
5
6
4
3
2
7
8
9
12
10
11
6
Characteristics of Probability Distribution

• Since we dont know the actual outcome (loss) in
  advance, we form an expectation about it. (For a
  loss distribution, the expected value is the
  expected loss)
• Expected Value of X ?PiXi
• Where
• Xi ith outcome of X (say, amount of loss)
• Pi probability of ith outcome of X

7
Characteristics of Probability Distribution

• Variance tells us about the likelihood and
  magnitude by which a particular outcome will
  differ from the expected value
• Variance of X ?Pi(Xi ?)2
• Where
• ? expected loss (from before)
• Pi, Xi are same as before

8
Characteristics of Probability Distribution

• Variance is expected deviation from the mean,
  squared
• Measure of dispersion
• Since Variance squares the unit of measure, we
  usually take the square root of the variance,
  which is the Standard Deviation
• Standard Deviation of X ?Pi(Xi ?)21/2

9
Example
10
Example

• Expected Loss .33 x 0 .34 x 500 .33 x 1,000
  500
• Variance .33(0 500)2 .34(500 500) 2
  .33(1,000 500) 2
• .33 x 250,000 .34 x 0 .33 x 250,000
• 165,000
• SD (165,000)1/2 406.2

11
Distribution Examples

• Consider the following distributions
• Normal bell shaped symmetric
• Normal having
• Same Standard Deviation, different Means
• Same Mean, different Standard Deviations
• Different Standard Deviations and Means
• Skewed distributions

12
Same Standard Deviation, Different Means
m2
m1
13
Same Mean, Different Standard Deviations
A
B
m1
14
Different Means and Standard Deviations
A
B
15
Skewed Distribution(Example Losses)
16
Population v. Sample

• We generally do not know the population
  characteristics
• We draw conclusions about population based on
  sample
• We can calculate the sample mean and sample
  standard deviation
• Can discuss the distribution of the sample mean

17
Risk Pooling

• Parties agree to evenly split the losses
• Reduces risk when losses are independent
• Does not change the expected loss
• Example Suppose Madison and Ryan each exposed to
  accident in coming year
• Outcome Probability
• 0 0.80
• 2,500 0.20

18
Without Pooling

• Expected loss .8 x 0 .2 x 2,500 500
• Standard deviation

19
With Pooling
20
With Pooling

• Expected loss (.64)(0) (.16)(2,500)
  (.16)(2,500) (.04)(2,500) 500

21
Distribution of Average Losses for Two Insured
Prob
.64
.32
.04
0
2,500
1,250
22
Loss Distribution w/4 insured
23
Distribution of Average Losses for Four Insured
Prob
.41
.10
.026
0
2,500
1,250
625
1,875
24
Distribution of Average Losses for 20 Insured
Prob
.2
500
25
Distribution With Without Pooling
With Pooling
Without Pooling
0
500
26
Law of Large Numbers

• As the sample size (of insured) increases, the
  average loss is likely to get very close to the
  expected loss (500 in this example)

27
Central Limit Theorem

• The distribution of sample means will be normally
  distributed (regardless of population
  distribution)
• The standard error of the sampling distribution
  declines as sample size increases

28
Central Limit Theorem

• Suppose SD of each participant in a pooling
  arrangement is initially 5,000
• With 10,000 participants the SD of losses will be
  50

29
Footnote

• The standard deviation of the average loss
  declines as the number of participants increases,
  but the standard deviation of the total losses
  for the group increases.
• The insured is concerned with SD of average loss,
  so that pooling does reduce risk.
• Pooling reduces risk that each participant has to
  bear.